Counting ( Enumerative Combinatorics) X. Zhang, Fordham Univ. 1 - - PowerPoint PPT Presentation

Counting 
 (Enumerative Combinatorics)

• X. Zhang, Fordham Univ.

1

Chance of winning ?

2

What’s the chances of winning New York Mega-

million Jackpot

“just pick 5 numbers from 1 to 56, plus a mega ball

number from 1 to 46, then you could win biggest potential Jackpot ever !”

If your 6-number combination matches winning 6-number

combination (5 winning numbers plus the Mega Ball), then you win First prize jackpot.

There are many possible ways to choose 6-number Only one of them is the winning combination… If each 6-number combination is equally likely to be

the winning combination …

Then the prob. of winning for any 6-number is 1/X

Counting

How many bits are need to represent 26 different

letters?

• How many different paths are there from a city to

another, giving the road map?

3

Counting rule #1: just count it

4

If you can count directly the number of

• utcomes, just count them.

For example:

How many ways are there to select an English

letter ?

26 as there are 26 English letters

How many three digits integers are there ?

These are integers that have value ranging from 100 to

999.

How many integers are there from 100 to 999 ?

999-100+1=900

Example of first rule

5

• How many integers lies within the range of 1 and 782

inclusive ?

• 782, we just know this !
• How many integers lies within the range of 12 and

782 inclusive ?

• Well, from 1 to 782, there are 782 integers
• Among them, there are 11 number within range from 1 to

11.

• So, we have 782-(12-1)=782-12+1 numbers between 12

and 782

Quick Exercise

6

So the number of integers between two integers, S

(smaller number) and L (larger number) is: L-S+1

How many integers are there in the range 123 to 928

inclusive ?

• How many ways are there to choose a number within

the range of 12 to 23, inclusive ?

A little more complex problems

7

How many possible license plates are available for

NY state ?

3 letters followed by 4 digits (repetition allowed)

How many 5 digits odd numbers if no digits can be

repeated ?

How many ways are there to seat 10 guests in a

table?

How many possible outcomes are there if draw 2

cards from a deck of cards ?

Key: all above problems ask about # of combinations/

arrangements of people/digits/letters/…

How to count ?

8

Count in a systematical way to avoid double-counting

• r miss counting

Ex: to count num. of students present …

First count students on first row, second row, … First count girls, then count boys

How to count (2)?

9

Count in a systematical way to avoid double-counting

• r miss counting

Ex: to buy a pair of jeans …

Styles available: standard fit, loose fit, boot fit and slim fit Colors available: blue, black How many ways can you select a pair of jeans ?

Use Table to organize counting

10

Fix color first, and vary styles

Table is a nature solution

• What if we can also choose size, Medium, Small
• r Large?

3D table ?

Selection/Decision tree

11

style color color color color Node: a feature/variable Branch: a possible selection for the feature Leaf: a configuration/combination

Let’s try an example

12

Enumerate all 3-letter words formed using

letters from word “cat”

assuming each letter is used once.

How would you do that ?

Choose a letter to put in 1st position, 2nd and 3rd

position

Exercises

13

• Use a tree to find all possible ways to buy a car
• Color can be any from {Red, Blue, Silver, Black}
• Interior can be either leather or fiber
• Engine can be either 4 cylinder or 6 cylinder
• How many different outcomes are there for a “best of

3” tennis match between player A and B?

• Whoever wins 2 games win the match…

Terminology

14

When buying a pair of jean, one can choose

style and color

We call style and color features/variables For each feature, there is a set of possible

choices/options

For “style”, the set of options is {standard, loose, boot,

slim}

For “color”, the set of options is {blue,black}

Each configuration, i.e., standard-blue, is called

an outcome/possibility

Outline on Counting

15

Just count it Organize counting: table, trees Multiplication rule Permutation Combination Addition rule, Generalized addition rule Exercises

Counting rule #2: multiplication rule

16

• If we have two features/decisions C1

and C2

• C1 has n1 possible outcomes/options
• C2 has n2 possible outcomes/options
• Then total number of outcomes is n1*n2
• In general, if we have k decisions to

make:

• C1 has n1 possible options
• …
• Ck has nk possible options
• then the total number of outcomes is

n1*n2*…*nk.

• “AND rule”:
• You must make all the decisions…
• i.e., C1, C2 , …, Ck must all occur

C1 C2 C2 C2 … … … … n1 n2 n2

Jean Example

17

• Problem Statement
• Two decisions to make: C1=Chossing style, C2=choosing

color

• Options for C1 are {standard fit, loose fit, boot fit, slim fit},

n1=4

• Options for C2 are {black, blue}, n2=2
• To choose a jean, one must choose a style and

choose a color

• C1 and C2 must both occur, use multiplication rule
• So the total # of outcomes is n1*n2=4*2=8.

Coin flipping

18

• Flip a coin twice and record the outcome (head or

tail) for each flip. How many possible outcomes are there ?

• Problem statement:
• Two steps for the experiment, C1= “first flip”,

C2=“second flip”

• Possible outcomes for C1 is {H, T}, n1=2
• Possible outcomes for C2 is {H,T}, n2=2
• C1 occurs and C2 occurs: total # of outcomes is n1*n2=4

License Plates

19

Suppose license plates starts with two different

letters, followed by 4 letters or numbers (which can be the same). How many possible license plates ?

Steps to choose a license plage:

Pick two different letters AND pick 4 letters/numbers. C1: Pick a letter C2: Pick a letter different from the first C3,C4,C5,C6: Repeat for 4 times: pick a number or letter

Total # of possibilities:

26*25*36*36*36*36 = 1091750400

Note: the num. of options for a feature/variable

might be affected by previous features

Exercises:

20

In a car racing game, you can choose from 4 difficulty

level, 3 different terrains, and 5 different cars, how many different ways can you choose to play the game ?

How many ways can you arrange 10 different

numbers (i.e., put them in a sequence)?

Relation to other topics

21

It might feel like that we are topics-hopping

Set, logic, function, relation …

Counting:

What is being counted ?

A finite set, i.e., we are evaluate some set’s cardinality when we

tackle a counting problem

How to count ?

So rules about set cardinality apply ! Inclusion/exclusion principle Power set cardinality Cartisian set cardinality

Learn new things by
 reviewing old…

Sets cardinality: number of elements in set

|AxB| = |A| x |B| The number of diff. ways to pair elements in A with

elements in B, i.e., |AxB|, equals to |A| x |B|

Example

A={standard, loose, boot}, the set of styles B={blue, black}, the set of colors AxB= {(standard, blue), (standard, black), (loose, blue),

(loose, black), (boot, blue), (boot, black)}, the set of different jeans

|AxB|: # of different jeans we can form by choosing from

A the style, and from B the color

22

Let’s look at more examples…

23

Seating problem

24

How many different ways are there to seat 5

children in a row of 5 seats?

Pick a child to sit on first chair Pick a child to sit on second chair Pick a child to sit on third chair …

The outcome can be represented as an ordered list: e.g.

Alice, Peter, Bob, Cathy, Kim

By multiplication rule: there are 5*4*3*2*1=120

different ways to sit them.

Note, “Pick a chair for 1st child” etc. also works

Job assignment problem

25

How many ways to assign 5 diff. jobs to 10

volunteers, assuming each person takes at most one job, and one job assigned to one person ?

Pick one person to assign to first job: 10 options Pick one person to assign to second job: 9 options Pick one person to assign to third job: 8 options … In total, there are 10*9*8*7*6 different ways to go

about the job assignments.

Permutation

26

Some counting problems are similar

How many ways are there to arrange 6 kids in a line ? How many ways to assign 5 jobs to 10 volunteers,

assuming each person takes at most one job, and one job assigned to one person ?

How many different poker hands are possible, i.e.

drawing five cards from a deck of card where order matters ?

Permutation

27

A permutation of objects is an arrangement where

• rder/position matters.

Note: “arrangement” implies each object cannot be picked

more than once.

Seating of children

Positions matters: Alice, Peter, Bob, Cathy, Kim is different from

Peter, Bob, Cathy, Kim, Alice

Job assignment: choose 5 people out of 10 and arrange

them (to 5 jobs)

Select a president, VP and secretary from a club

Permutations

28

Generally, consider choosing r objects out of a total

• f n objects, and arrange them in r positions.

… 1 2 3 r r-1

n objects (n gifts) r positions (r behaving Children)

Counting Permutations

29

Let P(n,r) be the number of permutations of r

items chosen from a total of n items, where r≤n

n objects and r positions

Pick an object to put in 1st position, # of ways: Pick an object to put in 2nd position, # of ways: Pick an object to put in 3rd position, # of ways: … Pick an object to put in r-th position, # of ways: By multiplication rule,

) 1 )...( 2 ( ) 1 ( ) , ( + − − ⋅ − ⋅ = r n n n n r n P

n n-1 n-2 n-(r-1)

Note: factorial

30

n! stands for “n factorial”, where n is positive integers,

is defined as

• Now

1 2 3 ... ) 1 ( ! ⋅ ⋅ ⋅ − ⋅ = n n n

)! ( ! 1 2 )... ( 1 2 )... ( ) 1 )...( 1 ( ) 1 )...( 1 ( ) , ( r n n r n r n r n n n r n n n r n P − = ⋅ − ⋅ − ⋅ + − − ⋅ = + − − ⋅ =

Examples

31

How many five letter words can we form using

distinct letters from set {a,b,c,d,e,f,g,h} ?

It’s a permutation problem, as the order matters and each

• bject (letter) can be used at most once.

P(8,5)

Examples

32

How many ways can one select a president, vice

president and a secretary from a class of 28 people, assuming each student takes at most one position ?

A permutation of 3 people selecting from 28 people:

P(28,3)=28*27*26

Exercises

33

What does P(10,2) stand for ? Calculate P(10,2).

• How about P(12,12)?
• How many 5 digits numbers are there where no digits

are repeated and 0 is not used ?

Examples: die rolling

34

If we roll a six-sided die three times and record results

as an ordered list of length 3

How many possible outcomes are there ?

6*6*6=216

How many possible outcomes have different results for

each roll ?

6*5*4

How many possible outcomes do not contain 1 ?

5*5*5=125

Combinations

35

Many selection problems do not care about

position/order

from a committee of 3 from a club of 24 people Santa select 8 million toys from store Buy three different fruits

Combination problem: select r objects from a set

• f n distinct objects, where order does not

matter.

Combination formula

36

C(n,r): number of combinations of r objects chosen

from n distinct objects (n>=r)

Ex: ways to buy 3 different fruits, choosing from apple,

• range, banana, grape, kiwi: C(5,3)

Ex: ways to form a committee of two people from a

group of 24 people: C(24,2)

Ex: Number of subsets of {1,2,3,4} that has two

elements: C(4,2)

Next: derive formula for C(n,r)

Deriving Combination formula

37

How many ways are there to form a committee of 2 for a

group of 24 people ?

Order of selection doesn’t matter

Let’s try to count:

There are 24 ways to select a first member And 23 ways to select the second member So there are 24*23=P(24,2) ways to select two peoples in

sequence

In above counting, each two people combination is

counted twice

e.g., For combination of Alice and Bob, we counted twice: (Alice,

Bob) and (Bob, Alice).

To delete overcounting

P(24,2)/2

General formula

38

when selecting r items out of n distinct items

If order of selection matters, there are P(n,r) ways For each combination (set) of r items, they have been

counted many times, as they can be selected in different orders:

For r items, there are P(r,r) different possible selection order

e.g., {Alice, Bob} can be counted twice: (Alice, Bob) and (Bob, Alice).

(if r=2)

Therefore, each set of r items are counted P(r,r) times.

The # of combinations is:

• )!

( ! ! )! /( ! )! /( ! ) , ( ) , ( ) , ( r n r n r r r r n n r r P r n P r n C − = − − = =

A few exercise with C(n,r)

39

• Calculate C(7,3)
• What is C(n,n) ? How about C(n,0)?
• Show C(n,r)=C(n,n-r).

)! ( ! ! ) , ( r n r n r n C − = 35 1 2 3 5 6 7 1 2 3 1 2 3 4 1 2 3 4 5 6 7 ! 3 )! 3 7 ( ! 7 ) 3 , 7 ( = ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = − = C

Committee Forming

40

How many different committees of size 7 can

be formed out of 20-person office ?

C(20,7) Three members (Mary, Sue and Tom) are carpooling.

How many committees meet following requirement ?

All three of them are on committee: None of them are on the committee:

C(20-7,4) C(20-7,7)

41

Outline on Counting

Just count it Organize counting: table, trees Multiplication rule Permutation Combination Addition rule, Generalized addition rule Exercises

Set Related Example

42

How many subsets of {1,2,3,4,5,6} have 3 elements ?

C(6,3)

How many subsets of {1,2,3,4,5,6} have an odd

number of elements ?

Either the subset has 1, or 3, or 5 elements. C(6,1)+C(6,3)+C(6,5)

Knapsack Problem

43

• There are n objects
• The i-th object has weight wi, and value

vi

• You want to choose objects to take

away, how many possible ways are possible ?

• 2*2*…*2=2n
• C(n,0)+C(n,1)+…+C(n,n)
• Knapsack problem:
• You can only carry W pound stuff
• What shall you choose to maximize the

value ?

Addition Rule

44

If the events/outcomes that we count can be

decomposed into k cases C1, C2, …, Ck, each having n1, n2, … nk, possible outcomes respectively,

(either C1 occurs, or C2 occurs, or C3 occurs, …. or Ck

• ccurs)

Then the total number of outcomes is n1+n2+…

+nk .

C1 C2 C3 C4

Key to Addition Rule

45

Decompose what you are

counting into simpler, easier to count scenarios, C1, C2, …, Ck

Count each scenario separately,

n1,n2,…,nk

Add the number together, n1+n2+

…+nk

C1 C3 C4 C2

Examples: die rolling

46

If we roll a six-sided die three times and record results

as an ordered list of length 3

How many of the possible outcomes contain exactly one 1,

e.g. 1,3,2 or, 3,2,1, or 5,1,3 ?

Let’s try multiplication rule by analyzing what kind of outcomes

satisfy this ?

First roll: 6 possible outcomes Second roll: # of outcomes ?

If first roll is 1, second roll can be any number but 1 If first roll is not 1, second roll can be any number

Third roll: # of outcomes ??

Examples: die rolling

47

If we roll a six-sided die three times and record

results as an ordered list of length 3

how many of the possible outcomes contain exactly one

1 ?

Let’s try to consider three different possibilities:

The only 1 appears in first roll, C1 The only1 appears in second roll, C2 The only1 appears in third roll, C3

We get exactly one 1 if C1 occurs, or C2 occurs, or C3

• ccurs

Result: 5*5+5*5+5*5=75

Examples: die rolling

48

If we roll a six-sided die three times, how many of the

possible outcomes contain exactly one 1 ? Let’s try another approach :

First we select where 1 appears in the list

3 possible ways

Then we select outcome for the first of remaining positions

5 possible ways

Then we select outcome for the second of remaining positions

5 possible ways

Result: 3*5*5=75

Example: Number counting

49

How many positive integers less than 1,000

consists only of distinct digits from {1,3,7,9} ?

To make such integers, we either

Pick a digit from set {1,3,7,9} and get an one-digit

integer

Take 2 digits from set {1,3,7,9} and arrange them to

form a two-digit integer

permutation of length 2 with digits from {1,3,7,9}.

Take 3 digits from set {1,3,7,9} and arrange them to

form a 3-digit integer

a permutation of length 3 with digits from {1,3,7,9}.

Example: Number Counting

50

Use permutation formula for each scenario (event)

# of one digit number: P(4,1)=3 # of 2 digit number: P(4,2)=4*3=12 # of 3 digit number: P(4,3)=4*3*2=24

Use addition rule, i.e., “OR” rule

Total # of integers less than 1000 that consists of {1,3,7,9}:

3+12+24=39

51

Example: computer shipment

Suppose a shipment of 100 computers contains 4

defective ones, and we choose a sample of 6 computers to test.

How many different samples are possible ?

C(100,6)

How many ways are there to choose 6 computers if all four

defective computers are chosen?

C(4,4)*C(96,2)

How many ways are there to choose 6 computers if one or

more defective computers are chosen?

C(4,4)*C(96,2)+C(4,3)*C(96,3)+C(4,2)*C(96,4)+C(4,1)*C(96,5) C(100,6)-C(96,6)

Generalized addition rule

52

If we roll a six-sided die three times how many

• utcomes have exactly one 1 or exactly one 6 ?

How many have exactly one 1 ?

3*5*5

How many have exactly one 6 ?

3*5*5

Just add them together ?

Those have exactly one 1 and one 6 have been counted twice How many outcomes have exactly one 1 and one 6 ?

C(4,1)P(3,3)=4*3*2

Generalized addition rule

53

If we have two choices C1 and C2,

C1 has n1 possible outcomes, C2 has n2 possible outcomes, C1 and C2 both occurs has n3 possible outcomes

then total number of outcomes for C1 or C2

• ccurring is n1+n2-n3.

C1 C2 C3

Generalized addition rule

54

If we roll a six-sided die three times how many

• utcomes have exactly one 1 or exactly one 6 ?

3*5*5+3*5*5-3*2*4

Outcomes that have exactly one 1, such as (1,2,3), (1,3,6), (2,3,1) Outcomes that have exactly one 6, such as (2,3,6), (1,3,6), (1,1,6) Outcomes that have exactly one 1 and one 6, such as (1,2,6), (3,1,6)

Example

55

A class of 15 people are choosing 3 representatives,

how many possible ways to choose the representatives such that Alice or Bob is one of the three being chosen? Note that they can be both chosen.

Summary: Counting

56

• How to tackle a counting problem?

1.

Some problems are easy enough to just count it, by enumerating all possibilities.

2.

Otherwise, does multiplication rule apply, i.e., a sequence of decisions is involved, each with a certain number of options?

Summary: Counting

57

How to tackle a counting problem?

• 3. Otherwise, is it a permutation problem ?

Summary: Counting (cont’d)

58

• How to tackle a counting problem?
• 4. Is it a combination problem ?

Summary: Counting (cont’d)

59

• How to tackle a counting problem?

5.

Can we break up all possibilities into different situations/cases, and count each of them more easily?

Summary: Counting (cont’d)

60

• How to tackle a counting problem?
• Often you use multiple rules when solving a particular

problem.

• First step is hardest.
• Practice makes perfect.

Exercise

61

A class has 15 women and 10 men. How many

ways are there to:

choose one class member to take attendance? choose 2 people to clean the board? choose one person to take attendance and one to clean

the board?

choose one to take attendance and one to clean the

board if both jobs cannot be filled with people of same gender?

choose one to take attendance and one to clean the

board if both jobs must be filled with people of same gender?

Exercise

62

A Fordham Univ. club has 25 members of which 5 are

freshman, 5 are sophomores, 10 are juniors and 5 are seniors. How many ways are there to

Select a president if freshman is illegible to be president? Select two seniors to serve on College Council? Select 8 members to form a team so that each class is

represented by 2 team members?

Cards problems

63

A deck of cards contains 52 cards.

four suits: clubs, diamonds, hearts and spades thirteen denominations: 2, 3, 4, 5, 6, 7, 8, 9, 10, J(ack),

Q(ueen), K(ing), A(ce).

begin with a complete deck, cards dealt are not put back

into the deck

abbreviate a card using denomination and then suit, such

that 2H represents a 2 of Hearts.

How many different flush hands?

64

A poker player is dealt a hand of 5 cards from a

freshly mixed deck (order doesn’t matter).

How many ways can you draw a flush? Note: a flush

means that all five cards are of the same suit.

More Exercises

65

A poker player is dealt a hand of 5 cards from a

freshly mixed deck (order doesn’t matter).

How many different hands have 4 aces in them? How many different hands have 4 of a kind, i.e., you have

four cards that are the same denomination?

How many different hands have a royal flush (i.e., contains

an Ace, King, Queen, Jack and 10, all of the same suit)?

66

Shirt-buying Example*

A shopper is buying three shirts from a store that

stocks 9 different types of shirts. How many ways are there to do this, assuming the shopper is willing to buy more than one of the same shirt?

There are only the following possibilities,

She buys three of the same type: Or, she buys three different type of shirts: Or, she buy two of the same type shirts, and one shift of

another type:

Total number of ways: 9+C(9,3)+9*8

9 C(9,3) 9*8

Round table seating

67

How many ways are there to arrange four

children (A,B,C,D) to sit along a round table, suppose only relative position matters ?

• As only relative position matters, let’s first fix a child, A,

how many ways are there to seat B,C,D relatively to A?

P(3,3)

A B D C A C D B

Same seating

Some challenges

68

• In how many ways can four boys and four girls sit

around a round table if they must alternate boy- girl-boy-girl?

• Hints:

1.

fix a boy to stand at a position

2.

Arrange 3 other boys

3.

Arrange 4 girls

Some challenges

69

A bag has 32 balls – 8 each of orange, white, red and

• yellow. All balls of the same color are
• indistinguishable. A juggler randomly picks three balls

from the bag to juggle. How many possible groupings

• f balls are there?

Hint: cannot use combination formula, as balls are not all

distinct as balls of same color are indistinguishable